What happens if a baby is left at an orphanage without identification?

There aren't orphanages in the US. There are Safe haven laws where if someone didn't want their child, they could drop them off at a hospital or fire station with no questions asked. The child would then be placed in foster care and the court would start the process of freeing the child for adoption. The county is still responsible for "Due Diligence" in reuniting the child with their family, but in the case of a safe haven baby due diligence is really just taking out a small ad in the newspaper notifying the parents of the child's placement. Assuming no one comes forward, the child would be adopted in less than 2yrs. The baby would not be fingerprinted for future identification.